We love Turtle Nest Inn in Bodden Town. We get a 1 bedroom apartment with a full kitchen and a rental car is included in the apartment rental costs. They also have condos for rent. It's on the water and they have 2 pools on the property - and a nice little beach and good snorkeling right offshore. There are some good, casual, local places to eat nearby and supermarkets for groceries, but it's a more residential area and not a place with a lot of restaurants, bars, or activities.
TNI is centrally located so it isn't terribly far from anything, but it is not a good choice if you want to do a lot of diving because you will have to drive to meet the boats or drive to the shore dive sites and the rush hour traffic can be bad.
When we stay at TNI and we want to boat dive we usually drive East to Ocean Frontiers or Tortuga, it's about a 30 minute drive and the traffic isn't bad on that end of the island; but it's not convenient if you want to make daily early morning boat dives. Ocean Frontiers offers 2 tank afternoon dives so that might be a nice option for you.
When we want to shore dive we usually drive to the West Bay area, north of Seven Mile Beach to dive Turtle Reef at Macabuca or some of the other shore dive sites. Sunset House near George Town is the nearest shore dive to TNI. The TNI price is higher than you are looking for, but it's quite reasonable for GC and a very nice property IMO, see link below:
Cayman Hotels :: Turtle Nest Inn, Grand Cayman - Exotic, Affordable Beachfront holiday
TNI also manages a small, 1-bedroom cottage called Sea Spray Cottage, see the link below, and you would have access to the pools at the Inn, but it is more than you hope to spend, see below:
Turtle Nest Affiliates :: Turtle Nest Inn, Grand Cayman affiliates- Accommotaion Beachfront holiday
A lot depends on how much diving you want to do? If you plan to do a lot of diving then it is better to stay at a dive resort or at a place nearer to the dive ops because traffic on GC can be bad, especially during rush hours. If you stay in the 7MB area then the dive shops will pick you up, but the hotel and condo rates are the highest in that area of the island.
Take a look at Cobalt Coast, they have a few free-standing cottages or suites adjacent to the resort that have kitchens, but I don't know how much they cost to rent.
Most people that go to CC stay in rooms at the resort and buy the all-inclusive packages that includes diving, food, and accommodations. CC is located north of 7MB, up near Hell, and they do provide airport transfers but it would be nice to have a car for at least part of the time to explore the island. There is a shore dive onsite but it tends to get winded out in the winter months.
You could buy one of the cheaper CC packages that includes 2 boat dives per day, shore diving, and 2 meals a day and then spend some time in the afternoons exploring the island and checking out places to eat lunch; see the link below:
Cobalt Coast
You could also look at the Grand Caymanian Holiday Inn, they have some 1-bedroom suites with a full kitchen. They are on the water and have a pool (or pools?). DiveTech has an onsite dive shop there and they will transport you to their Lighthouse Point property to board boats to dive or to shore dive right there. The Grand Caymanian is in the West Bay area and they provide shuttles to 7MB, but it would be best to have a car.
Holiday Inn Resort Grand Cayman
While you are comparison shopping the rental costs on GC, make sure that you are comparing "apples to apples" because in addition to a 13% sales tax, there may be service charges and resort fees added on to the cost of rental properties (and I think that the Morritt's resort charges a fee for using electricity.)
At TNI there will be a 13% tax added to the listed rate but no additional charges.
The all-inclusive rates for Cobalt Coast include the taxes, service charges, airport transfers, meals, accommodations, and diving - the only extra cost is the dive op tips - and a rental car if you decide to get one.
I don't know how they calculate the charges at the Grand Caymanian Holiday Inn.
I don't think that you will find much in your price range, but there are options in various price ranges. Good luck!